Found problems: 3585
Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle?
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$.
Show that $A,X,Y$ are collinear.
Let $\triangle{ABC}$ be an acute-angled triangle and let $D$, $E$, $F$ be points on $BC$, $CA$, $AB$, respectively, such that \[\angle{AFE}=\angle{BFD}\mbox{,}\quad\angle{BDF}=\angle{CDE}\quad\mbox{and}\quad\angle{CED}=\angle{AEF}\mbox{.}\] Prove that $D$, $E$ and $F$ are the feet of the perpendiculars through $A$, $B$ and $C$ on $BC$, $CA$ and $AB$, respectively.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 2)[/i]
The triangle $ABC$ is such that $\angle BAC=30^{\circ},\angle ABC=45^{\circ}$. Prove that if $X$ lies on the ray $AC$, $Y$ lies on the ray $BC$ and $OX=BY$, where $O$ is the circumcentre of triangle $ABC$, then $S_{XY}$ passes through a fixed point.
[i]Emil Kolev [/i]
Given a concyclic quadrilateral $ABCD$ with circumcenter $O$. Let $E$ be the intersection of $AD$ and $BC$, while $F$ be the intersection of $AC$ and $BD$. A circle $w$ are tangent to $BD$ and $AC$ such that $F$ is the orthocenter of $\triangle QEP$ where $PQ$ is a diameter of $w$. Prove that $EO$ passes through the center of $w$.
Let $AEF$ be a triangle with $EF = 20$ and $AE = AF = 21$. Let $B$ and $D$ be points chosen on segments $AE$ and $AF,$ respectively, such that $BD$ is parallel to $EF.$ Point $C$ is chosen in the interior of triangle $AEF$ such that $ABCD$ is cyclic. If $BC = 3$ and $CD = 4,$ then the ratio of areas $\tfrac{[ABCD]}{[AEF]}$ can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.
In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$.
[i]Proposed by Iman Maghsoudi[/i]
For $\triangle ABC$ and its circumcircle $\omega$, draw the tangents at $B,C$ to $\omega$ meeting at $D$. Let the line $AD$ meet the circle with center $D$ and radius $DB$ at $E$ inside $\triangle ABC$. Let $F$ be the point on the extension of $EB$ and $G$ be the point on the segment $EC$ such that $\angle AFB=\angle AGE=\angle A$. Prove that the tangent at $A$ to the circumcircle of $\triangle AFG$ is parallel to $BC$.
[i]Proposed by 61plus[/i]
Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. Circle $\omega_A$ is externally tangent to $\omega$ and tangent to sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Let $r_A$ be the line $A_1A_2$. Define $r_B$ and $r_C$ in a similar fashion. Lines $r_A$, $r_B$ and $r_C$ determine a triangle $XYZ$. Prove that the incenter of $XYZ$, the circumcenter of $XYZ$ and $I$ are collinear.
Let $\omega_1, \omega_2$ be two circles with different radii, and let $H$ be the exsimilicenter of the two circles. A point X outside both circles is given. The tangents from $X$ to $\omega_1$ touch $\omega_1$ at $P, Q$, and the tangents from $X$ to $\omega_2$ touch $\omega_2$ at $R, S$. If $PR$ passes through $H$ and is not a common tangent line of $\omega_1, \omega_2$, prove that $QS$ also passes through $H$.
Two circles $\omega_1,\omega_2$ intersect at $A,B$. An arbitrary line through $B$ meets $\omega_1,\omega_2$ at $C,D$ respectively. The points $E,F$ are chosen on $\omega_1,\omega_2$ respectively so that $CE=CB,\ BD=DF$. Suppose that $BF$ meets $\omega_1$ at $P$, and $BE$ meets $\omega_2$ at $Q$. Prove that $A,P,Q$ are collinear.
[i]Proposed by Iman Maghsoudi[/i]
Let $ABCD$ be a Rhombus and let $w$ be it's incircle. Let $M$ be the midpoint of $AB$ the point $K$ is on $w$ and inside $ABCD$ such that $MK$ is tangent to $w$. Prove that $CDKM$ is cyclic.
Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$.
Prove that the lines $AP$ and $CS$ are parallel.
In an acute-angled triangle, one of the angles is $60^o$. Prove that the line passing through the center of the circumcircle and the intersection point of the medians of the triangle cuts off an equilateral triangle from it.
(A. Zaslavsky)
In $\triangle ABC$, $AB = 6$, $BC = 7$, and $CA = 8$. Point $D$ lies on $\overline{BC}$, and $\overline{AD}$ bisects $\angle BAC$. Point $E$ lies on $\overline{AC}$, and $\overline{BE}$ bisects $\angle ABC$. The bisectors intersect at $F$. What is the ratio $AF$ : $FD$?
[asy] pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C); draw(A--B--C--A--D^^B--E); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,NE); label("$E$",E,NW); label("$F$",F,1.5*N); [/asy]
$\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2$
Let $\omega$ be a circle and $C$ a point outside it; distinct points $A$ and $B$ are selected on $\omega$ so that $\overline{CA}$ and $\overline{CB}$ are tangent to $\omega$. Let $X$ be the reflection of $A$ across the point $B$, and denote by $\gamma$ the circumcircle of triangle $BXC$. Suppose $\gamma$ and $\omega$ meet at $D \neq B$ and line $CD$ intersects $\omega$ at $E \neq D$. Prove that line $EX$ is tangent to the circle $\gamma$.
[i]Proposed by David Stoner[/i]
A acute triangle $\triangle ABC$ has circumcenter $O$. The circumcircle of $OAB$, called $O_1$, and the circumcircle of $OAC$, called $O_2$, meets $BC$ again at $D ( \not=B )$ and $E ( \not= C )$ respectively. The perpendicular bisector of $BC$ hits $AC$ again at $F$. Prove that the circumcenter of $\triangle ADE$ lies on $AC$ if and only if the centers of $O_1, O_2$ and $F$ are colinear.
Quadrilateral $ABCD$ is inscribed in circle. Points $P$ and $Q$ lie respectively on rays $AB^{\rightarrow}$ and $AD^{\rightarrow}$ such that $AP = CD$, $AQ = BC$. Show that middle point of line segment $PQ$ lies on the line $AC$.
$ABCD$ is a cyclic quadrilateral and $AC$ intersects $BD$ at $E$. $M, N$ are the midpoints of $AB, CD$, respectively. $\odot(AMN)$ meets $\odot(ABCD)$ again at $P$. $\odot(CMN)$ meets $\odot(ABCD)$ again at $Q$. $\odot(PEQ)$ meets $BD$ again at $T$. Prove that $M,N,T$ are colinear.
[i]Proposed by chengbilly[/i]
Let $ABC$ be a scalene triangle with circumcircle $\Gamma$. Let $M$ be the midpoint of $BC$. A variable point $P$ is selected in the line segment $AM$. The circumcircles of triangles $BPM$ and $CPM$ intersect $\Gamma$ again at points $D$ and $E$, respectively. The lines $DP$ and $EP$ intersect (a second time) the circumcircles to triangles $CPM$ and $BPM$ at $X$ and $Y$, respectively. Prove that as $P$ varies, the circumcircle of $\triangle AXY$ passes through a fixed point $T$ distinct from $A$.
Given is a triangle $ABC$ with circumcentre $O$. The circumcircle of triangle $AOC$ intersects side $BC$ at $D$ and side $AB$ at $E$. Prove that the triangles $BDE$ and $AOC$ have circumradiuses of equal length.
Suppose that $ ABCD$ is a convex quadrilateral. Let $ F \equal{} AB\cap CD$, $ E \equal{} AD\cap BC$ and $ T \equal{} AC\cap BD$. Suppose that $ A,B,T,E$ lie on a circle which intersects with $ EF$ at $ P$. Prove that if $ M$ is midpoint of $ AB$, then $ \angle APM \equal{} \angle BPT$.
Let $ABC$ be a triangle and let $BB_1,CC_1$ be respectively the bisectors of $\angle{B},\angle{C}$ with $B_1$ on $AC$ and $C_1$ on $AB$, Let $E,F$ be the feet of perpendiculars drawn from $A$ onto $BB_1,CC_1$ respectively. Suppose $D$ is the point at which the incircle of $ABC$ touches $AB$. Prove that $AD=EF$
In a triangle $ABC$, $\angle ACB=60^{\circ}$. Points $D, E$ lie on segments $BC, AC$ respectively. Points $K, L$ are such that $ADK$ and $BEL$ are equlateral, $A$ and $L$ lie on opposite sides of $BE$, $B$ and $K$ lie on the opposite siedes of $AD$. Prove that
$$AE+BD=KL.$$
In acute-angled triangle $ABC$, altitude of $A$ meets $BC$ at $D$, and $M$ is midpoint of $AC$. Suppose that $X$ is a point such that $\measuredangle AXB = \measuredangle DXM =90^\circ$ (assume that $X$ and $C$ lie on opposite sides of the line $BM$). Show that $\measuredangle XMB = 2\measuredangle MBC$.Proposed by Davood Vakili